Rather than using disk-based Recovery, try Internet Recovery - after the startup sound, hold down the command, option and R keys whilst booting. You'll (eventually) see a globe on the screen and your Mac will boot from Apple's servers into Internet Recovery. You should be able to use Disk Utility now, although there's no guarantee that your disk will be salvageable.

Even if you use the Online Internet Recovery system there is no provision for backing up your Mac in any of the recovery systems. They are for reinstalling OS X or restoring your Mac from a backup you made prior to any hard drive problems and for doing certain maintenance operations using Disk Utility.

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